HAL Id: hal-01174771
https://hal.archives-ouvertes.fr/hal-01174771v3
Submitted on 8 Jun 2017
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
The magnetic Laplacian acting on discrete cusps
Sylvain Golénia, Francoise Truc
To cite this version:
Sylvain Golénia, Francoise Truc. The magnetic Laplacian acting on discrete cusps. Documenta
Mathematica, Universität Bielefeld, 2017, 22, pp.1709-1727. �10.25537/dm.2017v22.1709-1727�.
�hal-01174771v3�
THE MAGNETIC LAPLACIAN ACTING ON DISCRETE CUSPS
SYLVAIN GOL´ENIA AND FRANC¸ OISE TRUC
Abstract. We introduce the notion of discrete cusp for a weighted graph. In this context, we prove that the form-domain of the magnetic Laplacian and that of the non-magnetic Laplacian can be different. We establish the emptiness of the essential spectrum and compute the asymptotic of eigenvalues for the magnetic Laplacian.
1. Introduction
The spectral theory of discrete Laplacians on graphs has drawn a lot of attention for decades. The spectral analysis of the Laplacian associated to a graph is strongly related to the geometry of the graph. Moreover, graphs are discretized versions of manifolds. In [MoT, GM], it is shown that for a manifold with cusps, adding a magnetic field can drastically destroy the essential spectrum of the Laplacian. The aim of this article is to go along this line in a discrete setting.
We recall some standard definitions of graph theory. A graph is a triple G := (E, V, m), where V is a countable set (the vertices), E : V × V → R+ is symmetric,
and m : V → (0, ∞) is a weight. We say that G is simple if m = 1 and E : V × V → {0, 1}.
Given x, y ∈ V, we say that (x, y) is an edge (or x and y are neighbors) if E(x, y) > 0. We denote this relationship by x ∼ y and the set of neighbors of x by NG(x). We say that there is a loop at x ∈ V if E(x, x) > 0. A graph is connected
if for all x, y ∈ V, there exists a path γ joining x and y. Here, γ is a sequence x0, x1, ..., xn ∈ V such that x = x0, y = xn, and xj ∼ xj+1 for all 0 ≤ j ≤ n − 1.
In this case, we set |γ| := n. A graph G is locally finite if |NG(x)| is finite for all
x ∈ V. In the sequel, we assume that:
All graphs are locally finite, connected with no loops. We endow a graph G := (E, V, m) with the metric ρG defined by
ρG(x, y) := inf{|γ|, γ is a path joining x and y}.
The space of complex-valued functions acting on the set of vertices V is denoted by C(V) := {f : V → C}. Moreover, Cc(V) is the subspace of C(V) of functions with
finite support. We consider the Hilbert space ℓ2(V, m) := ( f ∈ C(V),X x∈V m(x)|f (x)|2< ∞ )
with the scalar product hf, gi :=Px∈Vm(x)f (x)g(x).
We equip G with a magnetic potential θ : V × V → R/2πZ such that we have θx,y := θ(x, y) = −θy,x and θ(x, y) := 0 if E(x, y) = 0. We define the Hermitian
form QG,θ(f ) := 1 2 X x,y∈V
E(x, y) f(x) − eiθx,yf (y) 2,
2010 Mathematics Subject Classification. 34L20, 47A10, 05C63, 47B25, 47A63, 81Q10. Key words and phrases. discrete magnetic Laplacian, locally finite graphs, eigenvalues, asymp-totic, form-domain.
for all f ∈ Cc(V). The associated magnetic Laplacian is the unique non-negative
self-adjoint operator ∆G,θ satisfying hf, ∆G,θf iℓ2(V,m)= QG,θ(f ), for all f ∈ Cc(V).
It is the Friedrichs extension of ∆G,θ|Cc(V), e.g., [CTT3, RS], where
(∆G,θf )(x) = 1
m(x) X
y∈V
E(x, y) f (x) − eiθx,yf (y),
for all f ∈ Cc(V). We set
degG(x) := 1 m(x) X y∈V E(x, y),
the degree of x ∈ V. We see easily that ∆G,θ≤ 2 degG(·) in the form sense, i.e.,
0 ≤ hf, ∆G,θf i ≤ hf, 2 degG(·)f i, for all f ∈ Cc(V).
(1)
Moreover, setting ˜δx(y) := m−1/2(x)δx,y for any x, y ∈ V, h˜δx, ∆G,θδ˜xi = degG(x),
so ∆G,θ is bounded if and only if supx∈VdegG(x) is finite, e.g. [KL, Go].
Another consequence of (1) is
Ddeg1/2G (·)⊂ D∆1/2G,θ, (2)
where Ddeg1/2G (·):=f ∈ ℓ2(V, m), deg
G(·)f ∈ ℓ2(V, m)
. However, the equality of the form-domains
Ddeg1/2G (·)= D∆1/2G,θ (3)
is wrong in general for a simple graph, see [Go, BGK]. In fact if θ = 0, (2) is equivalent to a sparseness condition and holds true for planar simple graphs, see [BGK]. We refer to [BGKLM] for a magnetic sparseness condition. On a general weighted graph, if (3) holds true,
σess(∆G,θ) = ∅ ⇔ (∆G,θ+ 1)−1 is compact ⇔ lim
|x|→∞degG(x) = ∞,
where |x| := ρG(x0, x) for a given x0 ∈ V. Note that the limit is independent of
the choice of x0. Besides if the latter is true and if the graph is sparse (simple and
planar for instance), [BGK] ensures the following asymptotic of eigenvalues, lim n→∞ λn(∆G,θ) λn degG(·) = 1, (4)
where λn(H) denotes the n-th eigenvalue, counted with multiplicity, of a self-adjoint
operator H, which is bounded from below.
The technique used in [BGK] does not apply when the graph is a discrete cusp (thin at infinity), see Definition 2.5. The aim of this article is to establish new behaviors for the asymptotic of eigenvalues for the magnetic Laplacian in that case, and also to prove that the form-domain of the non-magnetic Laplacian can be different from that of the magnetic Laplacian, see Theorem 2.14. We found the inspiration by mimicking the continuous case, which was studied in [MoT, GM].
Let us present a flavour of our results (in particular of Theorem 2.14) by intro-ducing the following specific example of discrete cusp :
Example 1.1. Let n ≥ 3 be an integer and consider G1:= (E1, V1, m1), where
V1:= N, m1(n) := exp(−n), and E1(n, n + 1) := exp(−(2n + 1)/2),
for all n ∈ N and G2:= (E2, V2, 1) a simple connected finite graph such that |V2| = n.
product G1×V2G2, given by: m(x, y) := m1(x), E ((x, y), (x′, y′)) := E 1(x, x′) × δy,y′+ δx,x′× E2(y, y′), θ ((x, y), (x′, y′)) := δ x,x′× θ2(y, y′),
for all x, x′ ∈ V1 and y, y′ ∈ V2. Then there exists a constant ν > 0 such that for
all κ ∈ R/νZ σess(∆G,κθ) = ∅ ⇔ D ∆1/2G,κθ= Ddeg1/2G (·)⇔ κ 6= 0 in R/νZ Moreover: 1) When κ 6= 0 in R/νZ, we have: lim λ→∞ Nλ(∆G,κθ) Nλ degG(·) = 1,
where Nλ(H) := dim ran1]−∞,λ](H) for a self-adjoint operator H.
2) When κ = 0 in R/νZ, the absolutely continuous part of the ∆G,κθ is
σac(∆G,κθ) =
h
e1/2+ e−1/2− 2, e1/2+ e−1/2+ 2i,
with multiplicity 1 and lim λ→∞ Nλ ∆G,κθPac,κ⊥ Nλ degG(·) =n − 1 n ,
where Pac,κ denotes the projection onto the a.c. part of ∆G,κθ.
We now describe heuristically the phenomenon. Compared with the first case, the constant (n − 1)/n that appears in the second case encodes the fact that a part of the wave packet diffuses. Moreover, switching on the magnetic field is not a gentle perturbation because the form domain of the operator is changed.
By Riemann-Lebesgue Theorem, the particle, which is localized in the a.c. part of the operator, escapes from every compact set. More precisely, for a finite subset X ⊂ V and all f ∈ D(∆G,0)
k1X(·)eit∆G,0Pac,0f k → 0, as t → ∞.
In the first case, when the magnetic potential is active, the spectrum of ∆G,κθ is
purely discrete. The particle cannot diffuse anymore. More precisely, for a finite subset X ⊂ V and an eigenvalue f of ∆G,κθ such that f |X 6= 0, there is c > 0 such
that: 1 T Z T 0 k1X(·)eit∆G,κθf k2dt → c, as T → ∞.
The particle is trapped by the magnetic field.
· · ·
Diffusion
Magnetic effect
Representation of a discrete cusp:
The magnetic field traps the particle by spinning it, whereas its absence lets the particle diffuse.
We now describe the structure of the paper. In Section 2.1, we recall some properties of the holonomy of a magnetic potential. In Section 2.2 we present our
main hypotheses and several notions of (weighted) product for graphs. We introduce the notion of discrete cusp and analyze it under the light of the radius of injectivity. Then in Section 2.3 we give a criteria concerning the absence of essential spectrum. Next, in Section 2.4, we refine the analysis and give our central theorem, a general statement for discrete cusps, computing the form domain and the asymptotic of eigenvalues. We finish the section by proving Theorem 1.1.
Notation: Ndenotes the set of non negative integers and N∗ that of the positive
integers. We denote by D(H) the domain of an operator H. Its (essential) spectrum is denoted by σ(H) (by σess(H)). We set δx,y equals 1 if and only if x = y and 0
otherwise and given a set X, 1X(x) equals 1 if x ∈ X and 0 otherwise.
Acknowledgments: We would like to thank Colette Ann´e, Michel Bonnefont, Yves Colin de Verdi`ere, Matthias Keller, and Sergiu Moroianu for useful discussions. SG and FT were partially supported by the ANR project GeRaSic (ANR-13-BS01-0007-01) and by SQFT (ANR-12-JS01-0008-01).
2. Main results
2.1. Holonomy of a magnetic potential. We recall some facts about the gauge theory of magnetic fields, see [CTT3, HS] for more details and also [LLPP] for a different point of view. We recall that a gauge transform U is the unitary map on ℓ2(V, m) defined by
(U f )(x) = uxf (x),
where (ux)x∈V is a sequence of complex numbers with |ux| ≡ 1 (we write ux= eiσx).
The map U acts on the quadratic forms QG,θ by U⋆(QG,θ)(f ) = QG,θ(U f ), for all
f ∈ Cc(V). The magnetic potential U⋆(θ) is defined by:
U⋆(Q
G,θ) = QG,U⋆(θ).
More explicitly, we get:
U⋆(θ)xy = θx,y+ σy− σx.
We turn to the definition of the flux of a magnetic potential, the Holonomy. Proposition 2.1. Let us denote by Z1(G) the space of cycles of G. It is is a free
Z−module with a basis of geometric cycles γ = (x0, x1) + (x1, x2) + . . . + (xN −1, xN) with, for i = 0, · · · , N − 1, E(xi, xi+1) 6= 0, and xN = x0. We define the holonomy
map Holθ: Z1(G) → R/2πZ, by
Holθ((x0, x1) + (x1, x2) + · · · + (xN, x0)) := θx0,x1+ · · · + θxN,x0.
Then
1) The map θ 7→ Holθ is surjective onto HomZ(Z1(G), R/2πZ).
2) Holθ1= Holθ2if and only if there exists a gauge transform U so that U
⋆(θ
2) = θ1.
In consequence Holθ1 = Holθ2 if and only if the magnetic Laplacians ∆G,θ1 and
∆G,θ2 are unitarily equivalent.
Lemma 2.2. Let G := (E, V, m) be a connected graph such that 1 ∈ ker ∆G,0. Let
θ be magnetic potential. Then ker ∆G,θ6= {0} if and only if Holθ= 0.
Remark 2.3. By construction of the Friedrichs extension, the domain of ∆G,0 is
given by D(∆G,0) = f ∈ ℓ 2(V, m), x 7→ 1 m(x) X y∈V
E(x, y)(f (x) − f (y)) ∈ ℓ2(V, m)
\ Cc(V) (k·k2 +QG,0(·))1/2 .
The hypothesis 1 ∈ ker ∆G,0 is trivially satisfied if G is a finite graph. In general, it
(∗) 1 belongs to the closure of Cc(V) with respect to the norm (k·k2+QG,0(·))1/2.
A sufficient condition to guarantee (∗) is that the following two conditions hold true: 1) G is of finite volume, i.e., such thatPx∈Vm(x) < ∞,
2) ∆G,0 is essentially self-adjoint on Cc(V).
Proof. If Holθ= 0 then ∆G,θ is unitarily equivalent to ∆G,0 by Proposition 2.1 and
1 ∈ ker(∆G,0) 6= {0} by hypothesis.
Conversely, let f 6= 0 with ∆G,θf = 0 and hence QG,θ(f ) = 0. This implies that
all terms in the expression of QG,θ(f ) vanish. In particular, if E(x, y) 6= 0 we have
(5) f (x) = eiθx,yf (y).
Assume that there is a cycle γ = (x0, x1, . . . , xN = x0), such that Holθ(γ) 6= 0.
Using (5), we obtain that
f (xi) = e−iHolθ(γ)f (xi) .
for all i = 0, . . . , N − 1. Therefore f |γ = 0. Then, since f 6= 0, there is x ∈ V such
that f (x) 6= 0. Using again (5) and by connectedness between x and γ, it yields that f (x) = 0. Contradiction. Therefore if there exists f ∈ ker (∆G,θ) \ {0} then
Holθ= 0.
We exhibit the following coupling constant effect.
Corollary 2.4. Let G := (E, V, m) be a connected graph of finite volume, i.e., such thatPx∈Vm(x) < ∞ and let θ be a magnetic potential such that Holθ6= 0. Assume
that the function 1 is in ker ∆G,θ. Then there is ν ∈ R such that
ker ∆G,λθ 6= {0} ⇔ λ = 0 in R/νZ.
Proof. Let Φ : (R, +) → (HomZ(Z1(G), R/2πZ), +) be defined by Φ(λ) := Holλθ.
It is a homomorphism of group. Hence its kernel is a subgroup of (R, +). In particular it is either dense with respect to the Euclidean norm or equal to νZ for some ν ∈ R, e.g., [Bou, Section V.1.1]. Suppose by contradiction that the kernel is dense. Since for any cycle γ of G, the map λ 7→ Holλθ(γ) is continuous from R to
R/2πZ, we infer that Holλθ(γ) = 0 for all λ ∈ R. Hence, Φ(λ) = 0 for all λ ∈ R. This is a contradiction with Holθ 6= 0. We conclude that there is ν ∈ R such that
ker(Φ) = νZ, i.e., using Proposition 2.1, that
{λ ∈ R, ker ∆G,λθ 6= {0}} = {λ ∈ R, Holλθ= 0} = νZ.
This ends the proof.
2.2. The setting. Given G1:= (E1, V1, m1) and G2:= (E2, V2, m2), the Cartesian
product of G1 by G2 is defined by G := (E, V, m), where V := V1× V2.
m(x, y) := m1(x) × m2(y),
E ((x, y), (x′, y′)) := E
1(x, x′) × δy,y′m2(y) + m1(x)δx,x′× E2(y, y′),
θ ((x, y), (x′, y′)) := θ
1(x, x′) × δy,y′+ δx,x′× θ2(y, y′),
We denote by G := G1× G2. This definition generalizes the unweighted Cartesian
product, e.g., [Ha]. It is used in several places in the literature, e.g., [Ch][Section 2.6] and in [BGKLM] for a generalization.
· · · ·
The graph of Z × Z/3Z
The terminology is motivated by the following decomposition: ∆G,θ = ∆G1,θ1⊗ 1 + 1 ⊗ ∆G2,θ2,
where ℓ2(V, m) ≃ ℓ2(V
1, m1) ⊗ ℓ2(V2, m2). The spectral theory of ∆G,θ is
well-understood since
eit∆G,θ = eit∆G1,θ1 ⊗ eit∆G2,θ2, for t ∈ R.
We refer to [RS][Section VIII.10] for an introduction to the tensor product of self-adjoint operators.
In this paper, we are motivated by a geometrical situation. A hyperbolic manifold of finite volume is the union of a compact part and of a cusp, e.g., [Th, Theorem 4.5.7]. The cusp part can be seen as the product of (1, ∞) × M , where (M, gM) is
a possibly disconnected Riemannian manifold, endowed with the metric, y−1(dy2+ gM).
On the cusp part, the infimum of the radius of injectivity is 0.
To analyze the Laplacian on this product one separates the variables and obtain a decomposition which is not of the type of a Cartesian product, e.g., [GM, Eq. (5.22)] for some details. We aim at mimicking this situation and introduce a modified Cartesian product. Given G1:= (E1, V1, m1) and G2:= (E2, V2, m2) and I ⊂ V2, we
define the product of G1by G2 through I by G := (E, V, m), where V := V1× V2and
m(x, y) := m1(x) × m2(y), E ((x, y), (x′, y′)) := E 1(x, x′) × δy,y′ P z∈Iδy,z + δx,x′× E2(y, y′), θ ((x, y), (x′, y′)) := θ 1(x, x′) × δy,y′+ δx,x′× θ2(y, y′),
for all x, x′ ∈ V1 and y, y′ ∈ V2. We denote G by G1×I G2. If I is empty, the
graph is disconnected and of no interest for our purpose. If |I| = 1, G1×IG2 is the
graph G1 decorated by G2, see [SA] for its spectral analysis in the unweighted case.
If I = V2 and m = 1, we notice that G1×IG2= G1× G2.
· · · ·
The graph of Z The graph of Z/3Z
· · · ·
The graph of Z ×IZ/3Z, with |I| = 1
· · · ·
The graph of Z ×IZ/3Z, with |I| = 2
· · · ·
Under the representation ℓ2(V, m) ≃ ℓ2(V 1, m1) ⊗ ℓ2(V2, m2), degG(·) = degG1(·) ⊗ 1I(·) m2(·)+ 1 m1(·) ⊗ degG2(·) (6) and ∆G,θ= ∆G1,θ1⊗ 1I(·) m2(·) + 1 m1(·) ⊗ ∆G2,θ2. (7)
If m is non-trivial, we stress that the Laplacian obtained with our product is usu-ally not unitarily equivalent to the Laplacian obtained with the Cartesian product. However, there is a potential V : V → R such that ∆G1×G2 is unitarily equivalent
to ∆G1×V2G2+ V (·), in ℓ
2(V, m).
Definition 2.5. Set G1 := (E1, V1, m1), G2 := (E2, V2, m2), and I ⊂ V2. We say
that G = G1×IG2 is a discrete cusp if the following hypotheses are satisfied:
(H1) m1(x) tend to 0 as |x| → ∞,
(H2) G2 is finite,
(H3) ∆G1,θ1 is bounded (or equivalently supx∈V1degG1(x) < ∞).
We now motivate the choice of the above hypotheses by discussing the radius of injectivity. We start by defining a different metric on V, this choice is motivated by the works of [CTT2] and [MiT] but it needs a small adaptation for our purpose. Definition 2.6. Given G := (E, V, m), the weighted length of an edge (x, y) ∈ E defined by:
LG (x, y):=
s
min m(x), m(y)
E(x, y) .
Given x, y ∈ V, we define the weighted distance from x to y with respect to this length by: ρLG(x, y) := inf γ |γ|−1 X i=0 LG γ(i), γ(i + 1),
where γ is a path joining x to y and with the convention that ρLG(x, x) := 0 for all
x ∈ V.
Remark 2.7. Since G is assumed connected, ρLG is a metric on V. In fact ρLG
belongs to the class of intrinsic metrics. We refer to [Ke] for a general definition, historical references, properties, and applications. However, since Propositions 2.9 and 2.10 do not hold in general with an arbitrary intrinsic metric, we stick to our specific choice of metric.
We turn to the definitions of the girth and of the weighted radius of injectivity. This is essentially a weighted version of the standard ones, e.g, [EGL].
Definition 2.8. Given G := (E, V, m), the girth at x ∈ V of G w.r.t. the weighted length LG is
girth(x) := inf{LG(γ), γ simple cycle of unweighted length ≥ 3 and containing x},
where simple cycle means a closed walk with no repetitions of vertices and edges allowed, other than the repetition of the starting and ending vertex. We use the convention that the girth is +∞ if there is no such cycle.
girth(G) := inf
x∈Vgirth(x).
The radius of injectivity (at x) of G with respect to LG is half the girth (at x). We
denote the radius of injectivity by rad(G) (at x by rad(x) respectively) Note that with this definition, the radius of injectivity of a tree is +∞.
Proposition 2.9. Given G1 := (E1, V1, m1) and G2 := (E2, V2, m2) and I ⊂ V2
Assume that G := G1×IG2 is a discrete cusp. We have:
1) rad(G1) > 0.
2) If rad(G2) < ∞, then rad(G) = 0.
Proof. (1) Assume that rad(G1) = 0. Then for all ε > 0, there is x ∼ y in V1 such
that LG1 (x, y)
< ε. In particular, we have degG1(x) > ε
−2 or deg
G1(y) > ε
−2.
This is in contradiction with (H3).
(2) Since rad(G2) < ∞, for all x ∈ V1, there is a pure cycle contained in {x} × V2.
Moreover, for all x ∈ V1 and a ∼ b in V2, since E(x, x) = 0, we have:
LG1×IG2 ((x, a), (x, b))
=pm1(x)LG2 (a, b)
By (H1) we obtain that rad(G) = 0.
In contrast with this result we see that under the same hypotheses, the Cartesian product is not small at infinity. More precisely, we have:
Proposition 2.10. Set G1 := (E1, V1, m1) and G2 := (E2, V2, m2). Assume that
(H1), (H2), and (H3) are satisfied. Then rad(G1× G2) > 0.
Proof. Assume that rad(G1× G2) = 0. For all ε > 0, there are x1 ∼ y1 in V1 and
x2∼ y2 in V2 such that ε > LG1×G2 ((x1, x2), (x1, y2)) = LG2 (x2, y2) or ε > LG1×G2 ((x1, x2), (y1, x2)) = LG1 (x1, y1) .
The first line is in contradiction with (H2) and the second line with (H3). 2.3. Absence of essential spectrum. We have a first result of absence of essential spectrum. We refer to [CTT3] for related results based on the non-triviality of Holθ
in the context of non-complete graphs. See also [BGKLM] for similar ideas. Proposition 2.11. Set G1:= (E1, V1, m1), G2:= (E2, V2, m2), and G := G1×IG2,
with |I| > 0. Assume that (H1), (H2), and Holθ2 6= 0 hold true. Then ∆G,θ has a
compact resolvent, and
Nλ m−11 (·) ⊗ ∆G2,θ2
≥ Nλ(∆G,θ), for all λ ≥ 0.
Proof. Note that
∆G,θ≥
1 m1(·)
⊗ ∆G2,θ2
in the form sense on Cc(V). Since (H1) and (H2) hold, Lemma 2.2 ensures that
0 is not in the spectrum of (∆G2,θ2). Hence the spectrum of the r.h.s. is purely
discrete. By the min-max Principle, e.g., [Go, RS], the operator ∆G,θhas a compact
resolvent.
2.4. The asymptotic of the eigenvalues. From now on, we focus on the case when the graph is a discrete cusp and aim at a more precise result. To start off, we give the key-stone of our approach:
Proposition 2.12. Set G1 := (E1, V1, m1), G2 := (E2, V2, m2), and I ⊂ V2
non-empty. Assume that G := G1×IG2 is a discrete cusp. We set
M := sup
x∈V1
degG1(x) × maxy∈V 2 (1/m2(y)) < ∞. (8) We have: 1 m1(·) ⊗ degG2(·) ≤ degG(·) ≤ 1 m1(·) ⊗ degG2(·) + M, (9) 1 m1(·) ⊗ ∆G2,θ2 ≤ ∆G,θ≤ 2M + 1 m1(·) ⊗ ∆G2,θ2, (10)
in the form sense on Cc(V).
Proof. Use (1), (6), and (7).
We work in the spirit of [Go, BGK, BGKLM] and compare the Laplacian directly with the degree.
Proposition 2.13. Set G1 := (E1, V1, m1), G2 := (E2, V2, m2), and I ⊂ V2
non-empty. Assume that G := G1×I G2 is a discrete cusp. Set M as in (8). We
have:
inf σ(∆G2,θ2)
maxy∈V2degG2(y)
degG(·) − M
≤ ∆G,θ≤ 2M + 2 degG(·),
(11)
in the form sense on Cc(V).
Moreover, assuming that inf σ(∆G2,θ2) > 0, then D(∆
1/2 G,θ) = D
deg1/2G (·). Fur-thermore, since lim|x|→∞degG(x) = ∞, ∆G,θ has a compact resolvent and
0 < inf σ(∆G2,θ2)
maxy∈V2degG2(y)
≤ lim inf n→∞ λn(∆G,θ) λn(degG(·)) ≤ lim sup n→∞ λn(∆G,θ) λn(degG(·)) ≤ 2. Proof. Use (10) and (1) to get
inf σ(∆G2,θ2)
maxy∈V2degG2(y)
1 m1(·) ⊗ degG2(·) ≤ ∆G,θ≤ 2M + 2 m1(·) ⊗ degG2(·),
Then apply (9) to obtain (11). Concerning the statement about the eigenvalue this follows from the standard consequences of the min-max Principle, e.g., [Go]. Here, trying to compare directly ∆G,θ to degG to get sharp results about
eigen-values is too optimistic because it is unclear how to obtain constants arbitrarily close to 1 in front of degG, as in [Go, BGK]. To obtain some sharp asymptotics
for the eigenvalues of ∆G,θ, as in (15), we will use directly (10) and analyze very
carefully the operator m−11 (·) ⊗ ∆G2,θ2.
Theorem 2.14. Set G1:= (E1, V1, m1), G2:= (E2, V2, m2), and I ⊂ V2non-empty.
Assume that G := G1×IG2 is a discrete cusp. We obtain that
D(∆1/2G,θ) = Dm−1/21 (·) ⊗ ∆ 1/2 G2,θ2 . (12) Moreover, we have:
1) ∆G,θ has a compact resolvent if and only if Holθ2 6= 0.
2) If Holθ2 6= 0, then D(∆1/2G,θ) = Ddeg1/2G (·) and lim n→∞ λn(∆G,θ) λn m−11 (·) ⊗ ∆G2,θ2 = 1. (13) Furthermore, setting M as in (8), Nλ−2M m−11 (·) ⊗ ∆G2,θ2 ≤ Nλ(∆G,θ) ≤ Nλ m−11 (·) ⊗ ∆G2,θ2 , (14) for all λ ≥ 0.
Proof. First note that (12) follows directly from (10). Denoting by {gi}i=1,..,|V2|
the eigenfunctions associated to the eigenvalues {λi}i=1,..,|V2|of ∆G2,θ2, where λj ≤
λj+1, we see that the eigenfunctions of m−11 (·) ⊗ ∆G2 are given by {δx⊗ gi}, where
x ∈ V1 and i = 1, .., |V2|. Then, using (H1), we observe that
σ m−11 (·) ⊗ ∆G2 = m−11 (V1) × {λ1, . . . , λ|V2|} = m −1 1 (V1) × {λ1, . . . , λ|V2|}. Besides, 0 ∈ σ m−11 (·) ⊗ ∆G2
if and only if 0 is an eigenvalue of m−11 (·) ⊗ ∆G2 of
Moreover, recalling (H1), we see that all the eigenvalues of m−11 (·) ⊗ ∆G2 which are
not 0 are of finite multiplicity. Therefore, m−11 (·) ⊗ ∆G2 has a compact resolvent if
and only if Holθ2 6= 0. Combining the latter and (10), the min-max Principle yields
the first point.
We turn to the second point and assume that Holθ2 6= 0. The equality of the
form-domains is given by (11). Taking in account (10), the min-max Principle
ensures the asymptotic behavior of λn and the inequalities (14).
Remark 2.15. In the case when Holθ2 = 0, for instance when θ2= 0, we see that
the form-domain is m−1/21 ⊗Pker(∆⊥ G2,θ2). In particular, the form-domain is not that
of degG(·). Indeed if the two form-domains are the same, the closed graph theorem
yields the existence of c1> 0 and c2> 0 so that
c1degG(·) − c2≤ m −1/2
1 ⊗ Pker(∆⊥ G2,θ2),
in the form sense on Cc(V). However, note that 0 ∈ σess
m−1/21 ⊗ P⊥
ker(∆G2,θ2)
, whereas deg(·) has a compact resolvent. This is a contradiction with the min-max Principle. We obtain:
D∆1/2G,θ= Ddeg1/2(·)⇔ Holθ2 6= 0
⇔ ∆G,θ has a compact resolvent.
In (13), we exhibit the behaviour of the eigenvalues in terms of an explicit and computable mean. We now aim at comparing the asymptotic with that of the degree, as in [Go, BGK]. The new phenomenon is that we are able to obtain a constant different from 1 in the asymptotic.
Corollary 2.16. Let G1 := (E1, V1, m1), G2 := (E2, V2, m2), and I ⊂ V2
non-empty such that G := G1×IG2 is a discrete cusp. Suppose that degG2 is constant
on V2 and take θ2 such that Holθ2 6= 0. Then, for all a ∈ [1, +∞[, there exists
e
G1:= ( eE1, V1, ˜m1) such that
1) eG := eG1×IG2 is a discrete cusp.
2) E1 and eE1 have the same zero set.
3) degGe1(x) ≤ degG1(x) for all x ∈ V1.
4) ∆G,θe is with compact resolvent, and
lim λ→∞ Nλ ∆G,θe Nλ degGe(·) = a. (15)
Proof. We choose em1 and eE1 later. We denote by {λi}i=1,...,|V2|the eigenvalues of
∆G2,θ2. Since Holθ26= 0, we have λi6= 0 for all i = 1, . . . , |V2|. This yields:
Nλ 1 e m1 (·) ⊗ ∆G2,θ2 = (x, i), λi e m1(x) ≤ λ = |V2| X i=1 1 e m1 [−1] 0, λ λi , where [−1] denotes the reciprocal image. On the other hand,
Nλ 1 e m1(·) ⊗ degG2 = |V2| × 1 e m1 [−1] 0, λ degG2 . Moreover, from (9) we get
Nλ−M( em−11 (·) ⊗ degG2) ≤ Nλ(degGe(·)) ≤ Nλ( em
−1
1 (·) ⊗ degG2),
(16)
Step 1: We first aim at a = 1 in (15). Thanks to Lemma 2.18, we choose em1and
e
E1 such that the three first points are satisfied and
x ∈ V1, 1 e m1(x) ≤ λ ∼ ln(λ), as λ → ∞, where ∼ stands for asymptotically equivalent. We obtain:
Nλ 1 e m1(·)⊗ ∆G2,θ2 Nλ 1 e m1(·)⊗ degG2 ∼ P|V2| i=1(ln(λ) − ln(λi)) |V2|(ln(λ) − ln(degG2)) → 1, as λ → ∞. (17)
and for all c ∈ R, Nλ−c 1 e m1(·) ⊗ degG2 ∼ |V2| ln(λ − c) ∼ |V2| ln(λ) ∼ Nλ 1 e m1(·) ⊗ degG2 , as λ → ∞. (18)
Combining the latter with (16), we infer that for all c ∈ R Nλ−c 1 e m1(·) ⊗ degG2 ∼ Nλ(degGe(·)), as λ → ∞. (19)
Using now (17), this yields that for all c ∈ R Nλ−c 1 e m1(·) ⊗ ∆G2,θ2 ∼ Nλ(degGe(·)), as λ → ∞. (20)
Finally recalling (14), we infer that Nλ ∆G,θe ∼ Nλ degGe(·) , as λ → ∞.
In other words, there are em1 and eE1 such that the three first points are satisfied
and such that (15) is satisfied with a = 1.
Step 2: We turn to the case a > 1 in (15). Given α > 0,. Thanks to Lemma 2.18, we choose em1 and eE1such that the three first points are satisfied and
x ∈ V1, 1 e m1(x) ≤ λ ∼ λα, as λ → ∞, We obtain: Nλ 1 e m1(·)⊗ ∆G2,θ2 Nλ 1 e m1(·)⊗ degG2 ∼ λ→∞ 1 |V2| |V2| X i=1 deg G2 λi α =: F (α). First note that
lim
α→1+F (α) = 1.
Next, the sum of the eigenvalues (counted with multiplicity) of ∆G2,θ2 is equal to
|V2| degG2. Therefore, there exists at least one eigenvalue λi, with 1 ≤ i ≤ |V2| so
that degG2> λi. In particular
lim
α→+∞F (α) = +∞.
Finally, by continuity of F , we obtain that for all a > 1 there is α > 1 such that F (α) = a. To conclude, repeating the end of step 1, we obtain that for all a > 1, there are em1and eE1such that the three first points are satisfied and such that (15)
is satisfied.
Remark 2.17. In [Go, BGK], the asymptotic in Nλ was not discussed since the
estimates that they obtain seem too weak to conclude. Being able to compute Nλ in
We have used the following lemma:
Lemma 2.18. Let G1:= (E1, V1, m1) be a graph satisfying (H1) and (H3) in
Def-inition 2.5 and let f : [1, +∞) → [1, +∞) be a continuous and strictly increasing function that tends to +∞ at +∞. There exists eG1:= ( eE1, V1, em1) such that
1) E and ˜E have the same zero set. 2) (H1) and (H3) are satisfied for eG1.
3) degGe1(x) ≤ degG1(x) for all x ∈ V1.
4) We have: x ∈ V1, 1 e m1(x) ≤ λ ∼ f (λ), as λ → ∞, where ∼ stands for asymptotically equivalent.
Proof. Without any loss of generality, one may suppose that f (1) = 1. Let φ : N∗→ V1 be a bijection. Set:
˜
m1(φ(n)) :=
1 f[−1](n),
where [−1] denotes the reciprocal image. Note that (H1) is satisfied. Moreover, x ∈ V1, 1 e m1(x) ≤ λ = |{n ∈N∗, n ≤ f (λ)}| = ⌊f (λ)⌋ + 1 ∼ f (λ), as λ → ∞. Finally, we set: e E1(x, y) := E1(x, y) min( ˜m1(x), ˜m1(y)) max(m1(x), m1(y)) .
The first point is clear. For (H3), note that degGe1(x) ≤ degG1(x) for all x ∈ V1.
We end this section by proving the results stated in the introduction. Proof of Theorem 1.1. Let us consider G1:= (E1, V1, m1), where
V1:= N, m1(n) := exp(−n), and E1(n, n + 1) := exp(−(2n + 1)/2),
for all n ∈ N and G2:= (E2, V2, 1) a simple connected finite graph such that |V2| = n.
Set G := G1×V2G2, θ1:= 0 and θ2such that Holθ2 6= 0.
In the spirit of [GM], we denote by Pκle the projection on ker(∆G2,κθ2) and by
Phe
κ is the projection on ker(∆G2,κθ2)
⊥. Here le stands for low energy and he for
high energy.
We have that ∆G,κθ:= ∆leG,κθ⊕ ∆heG,κθ, where
∆leG,κθ:= ∆G1,0⊗ P le κ, on (1 ⊗ Ple κ)ℓ2(V, m), and ∆heG,κθ:= ∆G1,0⊗ P he κ + 1 m1(·) ⊗ Pκhe∆G2,κθ2, on (1 ⊗ Pκhe)ℓ2(V, m).
By Lemma 2.2, Corollary 2.4, and Remark 2.15, there exists ν > 0 such that Pκle= 0 ⇔ Holκθ2 6= 0
⇔ κ 6= 0 in R/νZ ⇔ D∆1/2G,κθ= Ddeg1/2G (·).
The proof of Theorem 2.14 gives the first point. Assume that κ ∈ R/νZ. Let U : ℓ2(N, m
1) → ℓ2(N, 1) be the unitary map given by U f (n) :=
p
m1(n)f (n). We
see that:
where ∆N,0is related to the simple graph of N. By using for instance some Jacobi
matrices techniques, it is well-known that the essential spectrum of ∆le
G,κθ is purely
absolutely continuous and equal to
σac(∆leG,κθ) = [e1/2+ e−1/2− 2, e1/2+ e−1/2+ 2],
with multiplicity one, e.g., [We]. It has a unique eigenvalue and it is negative. We turn to the high energy part. Denote by {λi}i=1,...,n, with λi ≤ λi+1, the
eigenvalues of ∆G2,κθ2. Recall that λ1= 0 due to the fact that Holκθ2 = 0. By (10),
1 m1(·) ⊗ ∆G2,κθ2P he κ ≤ ∆G,κθ(1 ⊗ Pκhe) ≤ 2M + 1 m1(·) ⊗ ∆G2,κθ2P he κ .
Hence, ∆G,κθ(1 ⊗ Pκhe) has a compact resolvent and
Nλ−2M m−11 (·) ⊗ ∆G2,κθ2P he κ ≤ Nλ ∆G,κθ(1 ⊗ Pκhe) ≤ Nλ m−11 (·) ⊗ ∆G2,κθ2P he κ , for all λ ≥ 0. Finally:
Nλ(m1 1(·)⊗ ∆G2,κθ2P he κ ) Nλ 1 m1(·)⊗ degG2 ∼ Pn i=2ln(λ) − ln(λi) n(ln(λ) − ln(degG2)) →n − 1 n , as λ → ∞.
We conclude with (18) for a = 1.
References
[BGK] M.Bonnefont, S.Gol´enia, and M.Keller: Eigenvalue asymptotics for Schr¨odinger oper-ators on sparse graphs, Ann. Inst. Fourier (Grenoble) 65 (2015), no. 5, 1969–1998. [BGKLM] M.Bonnefont, M.Keller, S.Gol´enia, S.Liu, and F.M¨unch: Magnetic sparseness and
Schr¨odinger operators on graphs, in preparation.
[Bou] N.Bourbaki: El´ements de math´ematiques: topologie g´en´erale – Chapitre 5 `a 10, Diffu-sion CCLS (1974), ISBN: 2-903684 006-3
[CTT2] Y.Colin de Verdi`ere, N.Torki-Hamza, and F.Truc: Essential self-adjointness for combi-natorial Schr¨odinger operators II: metrically non complete graphs, Math. Phys. Anal. Geom. 14 (2011), no. 1, 21–38.
[CTT3] Y.Colin de Verdi`ere, N.Torki-Hamza, and F.Truc: Essential self-adjointness for com-binatorial Schr¨odinger operators III: Magnetic fields, Ann. Fac. Sci. Toulouse Math. (6) 20 (2011), no. 3, 599–611.
[Ch] F.R.K.Chung: Spectral graph theory, CBMS Regional Conference Series in Mathemat-ics, 92. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1997. xii+207 pp. ISBN: 0-8218-0315-8
[DM] J.Dodziuk and V.Matthai: Kato’s inequality and asymptotic spectral properties for discrete magnetic Laplacians. The ubiquitous heat kernel, Cont. Math. 398, Am. Math. Soc. (2006), 69–81.
[EGL] S.Evra, K.Golubev, and A.Lubotzky, Mixing properties and the chromatic number of Ramanujan complexes, Int. Math. Res. Not. IMRN 2015, no. 22, 11520–11548. [Go] S.Gol´enia: Hardy inequality and asymptotic eigenvalue distribution for discrete
Lapla-cians, J. Funct. Anal. 266 (2014), no. 5, 2662–2688.
[Ha] F.Harary: Graph Theory, Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park, Calif.-London 1969 ix+274 pp.
[GM] S.Gol´enia and S.Moroianu: Spectral analysis of magnetic Laplacians on conformally cusp manifolds, Ann. Henri Poincar´e 9 (2008), no. 1, 131–179.
[HS] Y.Higuchi and T.Shirai: Weak Bloch property for discrete magnetic Schr¨odinger oper-ators, Nagoya Math. J. 161 (2001), 127–154.
[Ke] M.Keller: Intrinsic metric on graphs: a survey, Mathematical technology of networks, 81–119, Springer Proc. Math. Stat., 128, Springer, Cham, 2015.
[KL] M.Keller and D.Lenz: Unbounded Laplacians on graphs: basic spectral properties and the heat equation, Math. Model. Nat. Phenom. 5, no. 4, (2010) 198–224.
[LLPP] C.Lange, S.Liu, N.Peyerimhoff, and O. Post: Frustration index and Cheeger inequal-ities for discrete and continuous magnetic Laplacians, Calc. Var. Partial Differential Equations 54, no. 4, (2015), 4165–4196.
[MiT] O.Milatovic and F.Truc: Self-adjoint extensions of discrete magnetic Schr¨odinger op-erators, Annales Henri Poincar´e, 15 (2014), 917–936.
[MoT] A.Morame and F.Truc: Magnetic bottles on geometrically finite hyperbolic surfaces, J. Geom. Phys. 59 (2009), no. 7, 1079–1085.
[RS] M.Reed and B.Simon: Methods of Modern Mathematical Physics, Tome I–IV: Anal-ysis of operators Academic Press.
[SA] J.Schenker and M.Aizenman: The creation of spectral gaps by graph decoration, Lett. Math. Phys. 53 (2000), no. 3, 253–262.
[Th] W.P.Thurston: Three-Dimensional Geometry and Topology - Volume 1, Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997.
[We] J.Weidmann: Zur Spektraltheorie von Sturm-Liouville-Operatoren, Math. Z. 98 1967 268–302.
Institut de Math´ematiques de Bordeaux, 351, cours de la Lib´eration F-33405 Talence cedex
E-mail address: sylvain.golenia@math.u-bordeaux.fr
Grenoble University, Institut Fourier, Unit´e mixte de recherche CNRS-UJF 5582, BP 74, 38402-Saint Martin d’H`eres Cedex, France
